An important problem in medical statistics is estimation of the so-called minimum clinically important difference (MCID) that assesses the practical as opposed to statistical significance of a treatment. In words, the MCID is the threshold on the diagnostic measure scale such that improvements beyond that level are associated with patients feeling better after the treatment; see, e.g., [1]} and the references therein.
To set the scene, let \(X \in \mathbb {R}\)  denote the patient's diagnostic measure, e.g., the pre-treatment minus post-treatment difference in blood pressure, and let \(Y \in \lbrace -1,+1\rbrace \)  denote the patient-reported indicator of whether they felt the treatment was effective, with “\(y=+1\) ” indicating effective. The quantity of interest, \(\theta \) , the MCID, is the cutoff on the \(X\)  scale such that the indicator \(1\lbrace X > \theta \rbrace \)  is most highly associated with \(Y\) . More precisely, the MCID is defined as
\( \theta = \arg \min _\vartheta P\lbrace Y \ne \text{sign}(X-\vartheta )\rbrace , \) 
